| L(s) = 1 | + 0.233·2-s − 3-s − 1.94·4-s + 2.94·5-s − 0.233·6-s + 7-s − 0.921·8-s + 9-s + 0.687·10-s − 3.62·11-s + 1.94·12-s + 0.233·14-s − 2.94·15-s + 3.67·16-s − 1.53·17-s + 0.233·18-s + 4.10·19-s − 5.73·20-s − 21-s − 0.845·22-s − 7.03·23-s + 0.921·24-s + 3.67·25-s − 27-s − 1.94·28-s − 3.79·29-s − 0.687·30-s + ⋯ |
| L(s) = 1 | + 0.165·2-s − 0.577·3-s − 0.972·4-s + 1.31·5-s − 0.0953·6-s + 0.377·7-s − 0.325·8-s + 0.333·9-s + 0.217·10-s − 1.09·11-s + 0.561·12-s + 0.0623·14-s − 0.760·15-s + 0.918·16-s − 0.371·17-s + 0.0550·18-s + 0.940·19-s − 1.28·20-s − 0.218·21-s − 0.180·22-s − 1.46·23-s + 0.188·24-s + 0.735·25-s − 0.192·27-s − 0.367·28-s − 0.703·29-s − 0.125·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.233T + 2T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 + 7.03T + 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 - 7.24T + 37T^{2} \) |
| 41 | \( 1 + 9.15T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4.84T + 59T^{2} \) |
| 61 | \( 1 - 2.77T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 - 7.25T + 83T^{2} \) |
| 89 | \( 1 + 0.636T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.132804240067227959267545556346, −7.60927580273657464368687400536, −6.36032801438433117282590566458, −5.82056521182998710470578862615, −5.16673711019638889835081162210, −4.67382022298730165817329914179, −3.56974162883042052878134715662, −2.41968367367793173768516873478, −1.42328077418340625364709405609, 0,
1.42328077418340625364709405609, 2.41968367367793173768516873478, 3.56974162883042052878134715662, 4.67382022298730165817329914179, 5.16673711019638889835081162210, 5.82056521182998710470578862615, 6.36032801438433117282590566458, 7.60927580273657464368687400536, 8.132804240067227959267545556346
